In this paper, we extend this result to include the full Kerr-NUT-(A)dS class of spacetimes [1]}. Our extension is possible because these spacetimes all possess a series of Killing objects (vectors and higher tensors), all related to the `principal tensor' of the spacetime ([2]}; for a recent review, see [3]}). The principal tensor, a non-degenerate closed conformal Killing-Yano 2-form, not only provides for separability due to the conserved quantities from the Killing objects (see [3]} and citations therein, as well as [5]}),
it also ensures the spacetime is algebraically special [6]}. Indeed, the existence of the principal tensor, together with the assumption that the metric is a vacuum solution to Einstein's equation, uniquely identifies the Kerr-NUT-(A)dS class [7]}.
